Broadly speaking plant and animals that colonised and established naturally are considered native. Those that csme to th UK with human assistance are deemed introductions.
Sycamores and rabbits were both brought by (probably) the Romans, and are considered introductions.
Recent natural arrivals are considered 'colonisers - storks gor example.
Once established and self-sustaining spevies are considered 'naturalised'.
Invasive Non Native Species tend to cause problem. Often because without their usual predators and diseases they can dominate and damage existing ecosystems.
Some introductions and, more usually, reintroductions are often seen as beneficial... beavers... but can be controversial.
There are other categories: native, denizens, endemics...
These should be objective terms but obviously humans tend to put value judgements on things. So colonising Asian hornets are "bad"...